Which of the following graphs could be the graph of the function f(x) = x4 + x3 – x2 – x?

For this case we have the following function:
 
 We can rewrite the function to identify the zeros of it.
 When rewriting the function factoring we have: 
 Therefore, the zeros of the function are:
 
 Thus, the graph that contains intersections on the x axis in the points mentioned, will be the graph of the function.
 
 See attached image.

The given function is ⇒ f(x) = x⁴ + x³ – x² – x

To graph the function, we need to find zeros, minimum and maximum points.

The zeros of the function can be obtained as following;

f(x) = x⁴ + x³ – x² – x = (x⁴ + x³) – (x² + x) = x³ (x+1) - x (x+1) = (x+1) (x³ - x)

∴ f(x) = (x+1) x ( x² - 1) = x (x+1) (x+1) (x-1) = x (x-1) (x+1)²

So, the zeros will be at x = 0 , 1 , -1

To find the minimum and maximum points, we need to get f'(x)

∴ f'(x) = 4x³ + 3x² - 2x - 1

solve for x when f'(x) = 0 using the calculator.

∴ x = -1 , -0.39 , 0.64

Making a table between x and f(x) with taking the zeros and minimum and maximum points into considerations.

So, the graph of f(x) will be as the attached figure.

Ithink the answer would be knew

What am i doing here. add?